60 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (12.13.1) In a topological vector space E, the absorbing balanced neighborhoods of'0 form a fundamental system of neighborhoods 0/0. For each XQ e E, the mapping Ai-> Ax0 of the scalar field into E is con- tinuous at the point 0; hence, for each neighborhood V of 0 in E, there exists a > 0 such that the relation |A| ^ a implies Ax0 e V. This shows that every neighborhood of 0 is absorbing. On the other hand, the continuity of the mapping (A, jt)h->/bt at the point (0, 0) implies that, for each neighborhood V of 0 in E, there exists a > 0 and a neighborhood W of 0 in E such that the relations |A| <£ a and x e W imply Ix e V. The union U of the sets AW, where |A| ^ a, is clearly a balanced neighborhood of 0 in E, and is contained in V. Hence the result. Let F be a vector subspace of a topological vector space E. Then the topology induced on F by the topology on E is compatible with the vector space structure of F. Whenever we consider a vector subspace F as a topologi- cal vector space, it is always the induced topology that is meant, unless the contrary is expressly stated. The proof of (5.4.1) applies without any change and shows that F is a vector subspace of E. The definition of a total subset of E is the same as in (5.4). On the quotient vector space E/F, the quotient topology (12.11) is com- patible with the vector space structure. For if n : E -> E/F is the canonical mapping, let (A0, x0) be any point of R x (E/F) (resp. C x (E/F)) and let x0 be any point of ;c0. Then every neighborhood of A0 x0 contains a neighborhood of the form 7r(V), where V is a neighborhood of A0 x0. There exists a neighbor- hood U of A0 in R (resp. C) and a neighborhood W of x0 in E such that the relations AeU and xeW imply AxeV; consequently the relations AeU and x e 7i(W) imply AJC e 7r(V), and the assertion is proved. Whenever we consider E/F as a topological vector space, it is always with the quotient topology, unless the contrary is expressly stated. The criteria for continuity in a product ((3.20.15) and (12.5)) show immediately that if EA, E2 are two topological vector spaces, the product topology on E! x E2 is compatible with the vector space structure of Ex x E2. The vector space E1 x E2, endowed with this topology, is called the product of the topological vector spaces Et and E2. Proposition (5.4.2) and the defini- tion of the topological direct sum of two subspaces are valid without modifica- tion for arbitrary topological vector spaces. If E is the direct sum of two subspaces F1? F2, and if pl : E -* Fj_ is the corresponding projection, then the kernel of pt is F2, and/?A therefore has a, metrizable group, K a compact normal sub-